Tikhonov regularization gained a wide popularity in the context of Statistical inverse problems. In this article, we investigate and analyze fractional Tikhonov regularization scheme, which is in fact one of many generalizations of Tikhonov regularization, in learning theory. We delve into the theoretical framework of the fractional scheme to prove the consistency of the algorithm, and the convergence analysis of the solution. Furthermore, optimal rate of convergence has been established under Hölder source condition. A discussion on the rate of convergence is pursued with well established schemes in literature. Numerical experiment carried out through the academic data corroborate our analysis.

Recently, Reddy and Pradeep (2023) have proposed a class of parameter choice strategies to choose the regularization parameter for the finite dimensional weighted Tikhonov regularization scheme. In this article, we explore the iterated weighted Tikhonov scheme in the finite dimensional context, discuss its convergence analysis and propose a class of parameter choice strategies to choose the regularization parameter. Furthermore we establish optimal rate of convergence as Oδj(α+1)j(α+1)+1 based on the proposed strategies. The scheme’s performance is illustrated through numerical experiments with an efficient finite dimensional approximation of an operator.

Klann and Ramlau [16] hypothesized fractional Tikhonov regularization as an interpolation between generalized inverse and Tikhonov regularization. In fact, fractional schemes can be viewed as a generalization of the Tikhonov scheme. One of the motives of this work is the major pitfall of the a priori parameter choice rule, which primarily relies on source conditions that are often unknown. It necessitates the need for advocating a data-driven approach (a posteriori choice strategy). We briefly overview fractional scheme in learning theory and propose a modified Engl type [9] discrepancy principle, thus integrating supervised learning into the field of inverse problems. In due course of the investigation, we effectively explored the relation between learning from examples and the inverse problems. We demonstrate the regularization properties and establish the convergence rate of this scheme. Finally, the theoretical results are corroborated using two well known examples in learning theory.

Regularization is a method for providing a stable approximate solution to ill-posed operator equations, and it involves the regularization parameter which plays an important role in the convergence of the method. In this article, we propose a class of a posteriori parameter choice rules for filter-based regularization methods and establish the optimal rate of convergence O(δνν+1) from the proposed rules. We study these methods along with the proposed parameter choice rules in the context of pseudo-differential operator equations as well as the analytic continuation problem. The numerical implementation of the pseudo-differential operator equation and analytic continuation problem is discussed.

A priori parameter choice rules are successful in verifying the convergence of the reconstructed solutions. However, they suffer from a major drawback of utilizing source conditions, which, in most cases, is unknown. This pitfall is circumvented by invoking a posteriori parameter choice rules. In this article, we propound and investigate an Engl-type discrepancy principle for the choice of the regularization parameter in the learning theory perspective and establish the convergence rate. The consistency of the algorithm is an easy consequence. Moreover, we provide some insightful discussion on the weighted parameter. From a practical point of view, we demonstrate our theoretical analysis through two well-studied academic examples in learning theory.

In this article, we explore the further study of filter based schemes along with parameter choice rules and then apply to pseudo-differential operator equations and the analytic continuation problem. As we know that the stability of the regularization schemes depends on the regularization parameter and here we derive an a posteriori parameter choice rule which is best among all other parameter choice rules; and this parameter choice rule minimizes an upper bound of the approximation error ‖xα,δ−x†‖. Numerical experiments are also provided to validate the proposed theory.

Recently, Reddy (2018) has studied a posteriori parameter choice rules for the weighted Tikhonov regularization scheme. In this article, we consider the finite dimensional weighted Tikhonov scheme and discuss the convergence analysis of the scheme. We introduce a class of parameter choice strategies to choose the regularization parameter in the finite dimensional weighted Tikhonov scheme; and derive the optimal rate of convergence O(δα+1α+2) for the scheme based on the proposed strategies. Results of numerical experiments are documented to illustrate the performance of the scheme with an efficient finite dimensional approximation of an operator.

Many problems in science and engineering can be modeled as singularly perturbed partial differential equations. Solutions to such problems are not generally continuous with respect to the perturbation parameter(s) and hence developing stable and efficient numerical schemes to singularly perturbed problems are always more interesting and mathematically challenging to researchers. In this paper, we propose a generalized regularization scheme as an alternate numerical scheme for solving singularly perturbed parabolic PDEs with higher-order derivatives multiplied by a small parameter. Since the involved operators are unbounded, first, we propose a general theoretical framework for unbounded operators with an a posteriori parameter choice rule for selecting a regularization parameter, and then we apply it to singularly perturbed problems. We numerically implement the proposed scheme and compare it with other traditional schemes. The study illustrates that the proposed scheme can be considered as an alternate and competent method for solving singularly perturbed parabolic problems.

Singularly perturbed partial differential equations occur in many practical problems. These equations are characterized mathematically by the presence of a small parameter ϵ multiplying to one or more of the highest derivatives in a partial differential equation. Finding stable approximate solutions to such problems is mathematically challenging and interesting due to the fact that the solution is not stable as ϵ goes to 0. Many numerical schemes have been explored in the literature for finding approximate solutions to such PDEs. However, in this paper, our attempt is to propose a regularized iterative scheme as an alternative approach for finding stable approximate solutions to singularly perturbed elliptic and parabolic PDEs. We propose a general theoretical frame work for unbounded operators and then apply it for solving both singularly perturbed parabolic and elliptic PDEs. Theoretical results are illustrated through numerical examples and compared with other standard schemes. Numerical investigation assert that the proposed scheme is a very competitive and an alternate approach for solving the singularly perturbed problems.

Fractional Tikhonov regularization (FTR) method was studied in the last few years for approximately solving ill-posed problems. In this study we consider the Schock-type discrepancy principle for choosing the regularization parameter in FTR and obtained the order optimal convergence rate. Numerical examples are provided in this study.

Regularization procedure involves the regularization parameter that plays a crucial role in the convergence analysis of the regularization scheme. Recently, Reddy (2017) has proposed two a posteriori parameter choice rules to choose the regularization parameter in the weighted Tikhonov regularization scheme. The primary purpose of this article is to introduce a class of parameter choice rules to choose the regularization parameter in the stationary iterated weighted Tikhonov (SIWT) regularization scheme and derive the optimal rate of convergence O(δ[Formula presented]) for a stationary iterated method based on these proposed rules. The numerical experiments support our theoretical results.

Computing steering control for an approximately controllable linear system for a given target state is an ill-posed problem. We use a weighted Tikhonov regularization method and compute the regularized control. It is proved that the target state corresponding to the regularized control is close to the actual state to be attained. We also obtained error estimates and convergence rates involved in the regularization procedure using both the a priori and a posteriori parameter choice rule. Theory is substantiated with numerical experiments.

In this paper, we consider a class of singularly perturbed elliptical problems with homogeneous boundary conditions. We propose an iterative method for solving such problems. Convergence analysis and error estimate are derived. The proposed methodology is validated with the numerical results, and is also compared with well-known Shishkin scheme. The study demonstrates that the proposed regularized scheme has an edge over traditional numerical schemes.

The well-known approach to solve the ill-posed problem is Tikhonov regularization scheme. But, the approximate solution of Tikhonov scheme may not contain all the details of the exact solution. To circumference this problem, weighted Tikhonov regularization has been introduced. In this article, we propose two a posteriori parameter choice rules to choose the regularization parameter for weighted Tikhonov regularization and establish the optimal rate of convergence O(δα+1α+2) for the scheme based on these proposed rules. The numerical results are documented to demonstrate the significance of the theoretical results.

In this paper, we consider a class of singularly perturbed elliptical problems with homogeneous boundary conditions. We consider a regularized iterative method for solving such problems. Convergence analysis and error estimate are derived. The regularization parameter is chosen according to an a priori strategy. We give numerical results to illustrate that the method is implementable compared with numerical methods such as Shishkin and finite element schemes. The study demonstrates that the iterated regularized scheme can be considered as an alternate method for solving singularly perturbed elliptical problems.

In this paper, we examine the applicability of a variant of iterative Tikhonov regularization for solving parabolic PDE with its highest order space derivative multiplied by a small parameter ϵ. The solution of the operator equation (Formula Presented.) is not uniformly convergent to the solution of the operator equation ∂u∂t+a(x,t)=f(x,t), when ϵ→0. Although many numerical techniques are employed in practice to tackle the problem, the discretization of the PDE often leads to ill-conditioned system and hence the perturbed parabolic operator equation become ill-posed. Since we are dealing with unbounded operators, first we discuss the general theory for unbounded operators for iterated regularization scheme and propose an a posteriori parameter choice rule for choosing a regularization parameter in the iterative scheme. We then apply these techniques in the context of perturbed parabolic problems. Finally, we implement our iterative scheme and compare with other basic existing schemes to assert the adaptability of the scheme as an alternate approach for solving the problem.

Abstract In this paper, we examine the applicability of a variant of Tikhonov regularization for parabolic PDE with its highest order space derivative multiplied by a small parameter ∈. The solution of the operator equation (Formula presented.) is not uniformly convergent to the solution of the operator equation (Formula presented.), when ∈ → 0. This violates one of the conditions of Hadamard well-posedness and hence the perturbed parabolic operator equation is ill-posed. Since the operator is unbounded, we first discuss the general theory for unbounded operators and propose an a posteriori parameter choice rule for choosing a regularization parameter. We then apply these techniques in the context of perturbed parabolic problems. Finally, we implement our regularized scheme and compare with other basic existing schemes to assert the adaptability of the scheme as an alternate approach for solving the problem.